Multipole vortex patch equilibria for active scalar equations

Abstract

We study how a general configuration of finitely-many point vortices, in a state of uniform rotation or translation with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. Using a technique first introduced by Hmidi and Mateu for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular, at which point classical techniques such as Lyapunov-Schmidt reduction can be used. Provided the point vortex equilibrium is non-degenerate in a natural sense, solutions can be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we prove the existence of various families of solutions with patches arranged in asymmetric pairs, regular polygons, body-centered polygons, and nested regular polygons. These configurations are degenerate due to their additional symmetries, but this difficulty can be overcome by integrating the appropriate symmetries into the formulation of the problem.

Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations

“…An analytical proof based on the bifurcation theory and complex analysis tools was devised by Burbea in [18] who proved the existence of m-fold (for any m ∈ N * ) symmetric V-states bifurcating from Rankine vortices with angular velocity Ω m := m−1 2m . More investigations on the V-states in different settings like the doubly-connected patches, vortex pairs, boundary effects or for different models has been implemented during the past decade by several authors, for more details we refer to [19,20,27,28,29,30,31,32,33,34,36,37,38,39,40]. Concerning (QGSW) λ there are a few results dealing with relative equilibria.…”

KAM theory for active scalar equations

“…The same technique was used to desingularize a spatial periodic distribution called Karman vortex street [44] and a similar study was performed for the Thomson polygon [45]. Very recently, a generic system of rotating point vortices was analyzed in [56] and a weak condition on the validity of the desingularisation through the contour dynamics has been found. 1.2.…”

Vortex rigid motion in quasi-geostrophic shallow-water equations

“…We emphasize that different studies around this subject have been recently investigated by several authors, we refer for instance to [13,14,15,18,22,27] and the references therein. The main contribution of this paper is to establish for (QGSW ) λ the existence of branches of bifurcation in the doubly-connected case, generalizing the result of [24].…”